In Parser.gd you have these lines which call the next() function:
func _input(event):
if Input.is_action_just_pressed(dialogue_next_input):
MSG.next()
As you can see the button for skipping the dialogue is called dialogue_next_input which can be changed in Project > Project Settings > Input Map > dialogue_next - by default it's [Space].
As for the voice sounds, you can add an export variable to characters, like this: export (AudioStream) var sound and then in start_message() in MSG_Box.gd and MSG_Bubble.gd add something like $SFX.stream = MSG.current_speaker.sound. That should work (haven't tested it though).
I hope that answers your questions :)